Application of fused lasso logistic regression to the study of corpus callosum thickness in early Alzheimer's disease

Abstract

We propose a fused lasso logistic regression to analyze callosal thickness profiles. The fused lasso regression imposes penalties on both the l₁-norm of the model coefficients and their successive differences, and finds only a small number of non-zero coefficients which are locally constant. An iterative method of solving logistic regression with fused lasso regularization is proposed to make this a practical procedure. In this study we analyzed callosal thickness profiles sampled at 100 equal intervals between the rostrum and the splenium. The method was applied to corpora callosa of elderly normal controls (NCs) and patients with very mild or mild Alzheimer’s disease (AD) from the Open Access Series of Imaging Studies (OASIS) database. We found specific locations in the genu and splenium of AD patients that are proportionally thinner than those of NCs. Callosal thickness in these regions combined with the Mini Mental State Examination scores differentiated AD from NC with 84% accuracy.

1. Introduction

Alzheimer’s disease (AD) is an age-related, irreversible neurodegenerative disease that develops gradually and results in episodic memory loss and behavioral and personality changes. There is great interest in finding neuroimaging based markers to predict the progression of AD, which could be very helpful in managing care for individual patients, and in AD research to develop interventions for altering the course of disease. Gray matter, especially the medial temporal lobe (MTL) and hippocampus, is a major focus in these studies (see Risacher and Saykin (2013) for a comprehensive review), but it is hard to reliably measure the MTL and hippocampus automatically. On the other hand, a number of studies have reported structural changes in the corpus callosum (CC) in AD (Di Paola et al., 2010; Frederiksen et al., 2011; Teipel et al., 2002; Wang et al., 2006), and there are fully automated procedures to measure the CC with good accuracy (Ardekani et al., 2012a). In this paper, we attempt to identify the regions of the CC whose thickness could differentiate patients with early AD from normal controls (NCs) and use them in a fused lasso logistic regression classifier for early AD.

Two major approaches have been used to localize the differences in the CC between two populations: (1) defining a small number of geometric regions and (2) measuring the thickness of the CC at a large number (50–100) of evenly spaced points along the midline and doing regression or factor analysis to determine locations of significant difference. In (1) there are a number of different geometric parcellations in use (Hampel et al., 2002; Weis et al., 1991; Witelson, 1989) and no general agreement on how many divisions are meaningful. Method (2) does not require a priori parcellations and can give much finer detail, but suffers from multiple comparisons and does not exploit the correlations expected between neighboring thickness. Factor analysis has been used to divide the CC into anatomically “meaningful” subdivisions and a post-hoc ANOVA-type analysis has been applied to these subdivisions (Denenberg et al., 1991b; Hallam et al., 2008; Kilian et al., 2007). However, factor analysis can be subjective in determining the number of regions and it is not reliable in determining subdivisions.

We explore the second approach to corpus callosum analysis by measuring its thickness profile from rostrum to splenium. We introduce a new statistical approach to analyze the thickness profile to find regional differences in the CC between two populations. To this end, we extend a logistic regression model by incorporating fused lasso regularization to select contiguous regions rather than individual thicknesses. This approach does not require a priori divisions of the CC, but is able to identify regionally specific differences in CC thickness between two populations. We demonstrate the proposed method to determine regional specificity of the CC thickness differences between NCs and patients with very mild or mild AD. For this purpose, we utilize images from the Open Access Series of Imaging Studies (OASIS) MRI database (Marcus et al., 2007).

2. Method

2.1. MRI data

The OASIS cross-sectional data set (Marcus et al., 2007) contains 3D MPRAGE MRI brain scans from 416 right-handed subjects. Of these, we include in this study the 98 healthy normal subjects aged 60 or above without dementia (CDR=0) and all 98 subjects aged 60 or above with very mild/mild AD (CDR=0.5 or 1). See Marcus et al. (2007) for details of subject demographics, inclusion/exclusion criteria, MRI acquisition protocol, and preprocessing steps. Table 1 summarizes the demographics of the subjects included in this paper.

Table 1.

Sample characteristics of the data set as given by Marcus et al. (2007). Values are mean ± SD (minimum - maximum). The range of MMSE score is between 0 (worst) and 30 (best).

	CDR=0	CDR=0.5	CDR=1
Number	98	70	28
Sex (female/male)	72/26	39/31	19/9
Age (years)	75.9 ± 9.0 (60–94)	76.4 ± 7.0 (63–92)	82 ± 5.7 (78–86)
MMSE score	29.0 ± 1.2 (25–30)	25.6 ± 3.5 (14–30)	21.7 ± 3.8 (15–29)

2.2. Corpus callosum segmentation

A fully automated method (Ardekani et al., 1997) is applied to find the mid-sagittal plane (MSP) of the MRI volumes to bring the head yaw and roll angles as close as possible to zero. In addition, the anterior commissure (AC) and posterior commissure (PC) were located on the MSP using a fully automated model-based method (Ardekani and Bachman, 2009). Using this information, the original MRI volume was re-sliced to obtain a single image of matrix size: 512 × 512 and pixel size: 0.5 × 0.5 mm² representing the true AC-PC aligned MSP. Using a priori information obtained from manually traced corpora callosa on this type of image, a rectangular CC search region was identified on the MSP. Finally, a multi-atlas model-based segmentation method (Aljabar et al., 2009; Ardekani et al., 2012a,b; Cabezas et al., 2011) using the Automatic Registration Toolbox (ART) non-linear registration algorithm (Ardekani et al., 2005; Klein et al., 2009) was used to locate the CC within the search region as shown in Figure 1(a). If necessary, small corrections to the detected CC were made manually using the ITK-SNAP software (Yushkevich et al., 2006) by an operator blinded to subject status. The final segmentation was represented as a binary image where pixels were assigned a value of 1 if they belonged to the CC and a value of 0 otherwise. The CC segmentation algorithm described above has been implemented in C++ and using message passing interface (MPI) parallel programming library. The software, yuki, is available publicly at www.nitrc.org/projects/art. The average processing time for a CC segmentation running yuki in parallel on 7 of the 8 existing Cores in a 2.4 GHz Dual Quad-Core Linux computer was less than 7 seconds. Running the program on a single Core in serial mode required approximately 28 seconds.

Figure 1.

(a) The outline shows the detected corpus callosum. The automatically detected anterior commissure (AC) and posterior commissure (PC) are shown by the plus signs. (b) An example CC thickness profile. Every tenth medial axis position ${\mathbf{m}}_i^{*}$ is indicated by a diamond. Numbering is 0 at the rostrum tip (r) and 100 at the inferior point of the splenium (s).

It must be emphasized that in the multi-atlas based approach that we have used, the non-linear registrations are applied to the CC atlases only, not to the test image being measured. The test image is treated as a fixed target to which floating atlases are matched. Therefore, the CC of the test image does not undergo any shape or size alternation due to the non-linear registration processes involved in the algorithm. The only alignment that is applied to the test image is a non-shape/size altering rigid-body transformation based on the detected MSP and AC and PC landmarks that brings the test image to a standardized orientation, as described above, before multi-atlas based segmentation is applied.

2.3. Corpus callosum thickness profile

The CC thickness profile is specified in terms of 99 non-zero thickness values at equally spaced intervals along the length of the CC. Our method for finding these values is similar to those of Clarke et al. (1989) and Denenberg et al. (1991a). More specifically, the thickness values are lengths of line segments that connect pairs of points on the upper and lower boundaries of the CC. The line segments are perpendicular to the medial axis of the CC and intersect the medial axis at equal intervals. The steps for finding the thickness values are as follows.

• (a)The tip of the rostrum r (point 0 in Figure 1(b)) and the inferior point of the splenium s (point 100 in Figure 1(b)) are located automatically in the segmented binary CC image. For this purpose, we use information based on the known CC orientation with respect the AC and PC landmarks.
• (b)Using the landmarks (rostrum tip and splenium bottom) found in step (a), the CC boundary is divided into two parts, an upper curve and a lower curve, 101 equally spaced points u_i are placed on the upper boundary and 101 equally spaced points l_i are located on the lower boundary, where i = 0, 1,…, 100, u₀ = l₀ = r, and u₁₀₀ = l₁₀₀ = s.
• (c)The mid-points m_i = (u_i + l_i)/2 between the corresponding upper and lower CC boundary pairs u_i and l_i are found to define 101 points on a medial axis of the CC.
• (d)A medial axis is reconstructed by linear interpolation between the m_i and smoothed. Then, a new set of 101 equally spaced points ${\mathbf{m}}_i^{*}$ are placed on the smoothed medial axis, such that ${\mathbf{m}}_0^{*}=\mathbf{r}$ and ${\mathbf{m}}_{100}^{*}=\mathbf{s}$.
• (e)The normal lines to the medial axis at points ${\mathbf{m}}_i^{*}$ (i = 1, 2,…, 99) are determined and their intersection points with the upper and lower boundaries of the CC are detected and denoted by ${\mathbf{u}}_i^{*}$ and ${\mathbf{l}}_i^{*}$, respectively.
• (f)Finally, the CC thickness profile is defined as x = {x_i} (i = 1, 2,…, 99) where x_i are the distances between points ${\mathbf{u}}_i^{*}$ and ${\mathbf{l}}_i^{*}$, that is, $x_i=\Vert {\mathbf{u}}_i^{*}-{\mathbf{l}}_i^{*}\Vert$.

The above algorithm makes two assumptions that may not be necessarily true. The first is that, as shown in Figure 1(b), the perpendicular line segments to the medial axis do not cross each other within the CC. The second assumption is that a perpendicular to the medial axis intersects each of the upper and lower boundaries exactly once. In occasional cases, these assumptions are violated especially in the genu-rostral area. To deal with this situation, the following two steps are added to the above algorithm.

• (g)If a line segment perpendicular to the medial axis at a point ${\mathbf{m}}_i^{*}$ crosses another line segment within the CC or if a line segment does not cross the upper and lower boundaries exactly once, delete the corresponding points ${\mathbf{u}}_i^{*}$ and ${\mathbf{l}}_i^{*}$.
• (h)Replace the missing ${\mathbf{u}}_i^{*}$ and ${\mathbf{l}}_i^{*}$ points by placing equi-distance points between the closest existing neighboring points. For example, if ${\mathbf{u}}_i^{*}$ were deleted in step (g), replace it with a point that bisects the arc on the upper boundary that joints ${\mathbf{u}}_{i-1}^{*}$ and ${\mathbf{u}}_{i+1}^{*}$. Extension to cases where more than two adjacent boundary points are missing is trivial. Once the boundary points are corrected, compute the thicknesses $x_i=\Vert {\mathbf{u}}_i^{*}-{\mathbf{l}}_i^{*}\Vert$ as before.

2.4. Logistics regression with fused lasso regularization

The logistic regression models the probability of an event of interest, denoted by p_i, using a univariate score of many predictive variables as p_i = exp(s_i)/{1 + exp(s_i)} where s_i is a weighted linear sum of predictive variables. The relationship between s_i and p_i is the well-known s-shaped curve. Often s_i is used as a classifier in the sense that we classify the i-th subject into the positive class (the class of an event of interest) if s_i ≥ 0, which is equivalent to p_i ≥ 1/2. Otherwise, the subject is classified into the normal group. The ℓ₁ regularization (lasso: least absolute shrinkage and selection operator) was introduced by Tibshirani (1996) to simultaneously select and estimate the model. The lasso method finds only a few non-zero coefficients (i.e., weights of the predictive variables that form s_i) and becomes a useful tool when the number of predictors is large but only a few of them are actually effective.

The classical lasso regression by Tibshirani (1996), which penalizes the ℓ₁ norm of the coefficients vector (the sum of absolute values of coefficients) of ordinary least squares, has many variants to incorporate additional structural information on covariates. For example, the elastic net by Zou and Hastie (2005) is designed to simultaneously select strongly correlated variables in the model. The group lasso by Yuan and Lin (2007) again jointly selects or de-selects a group of variables to explain a single categorical variable. The fused lasso regression by Tibshirani et al. (2005) assumes that the predictors are observed in an order and penalizes a convex combination of the ℓ₁ norm of differences of adjacent coefficients and the ℓ₁ norm of the coefficient vector itself. It is also known that, like the elastic net, the fused lasso regression selects correlated variables together but it further makes the coefficients to be locally equal to each other.

The callosal thicknesses are spatially correlated in the sense that the thickness at one point is correlated to thicknesses of its neighboring point, and we are interested in finding regions rather than individual thickness points that would differentiate between groups. For this reason, the fused lasso logistic regression (FLLR) is a suitable choice for the analysis of callosal thickness profiles, and results in locally homogeneous and spatially contiguous regions that would indicate the existence of the event of interest (e.g., a subject belonging to the AD group).

Suppose that we have n samples {(y_i, d_i, x_i), i = 1, 2,…, n} (n = 196 in the current study), where d_i = (d_i1, d_i2,…,d_is) is an s-dimensional covariate vector such as demographic information, x_i = (x_i1, x_i2,…,x_it) is a t-dimensional set of callosal thicknesses (t = 99 in the current study), and y_i is a binary variable indicating status of the i-th subject. In this paper, y_i = 1 if the i-th subject is in the AD group, otherwise 0. In order to find regional differences of the CC thickness between two populations controlling for covariates, we consider the logistic regression model:

$$g(p_i)=\text{log}\frac{p_i}{1-p_i}={\mathbf{d}}_i\gamma +{\mathbf{x}}_i\beta ,$$

where p_i = Pr(y_i = 1), γ = (γ₁,…, γ_s)^T and β = (β₁,…, β_t)^T. The likelihood function of (γ, β) is given by

$$L(\gamma ,\beta ;\mathbf{y},\mathbf{d},\mathbf{x})=\prod _{i=1}^n{p}_i^{y_i}{(1-p_i)}^{1-y_i}=\prod _{i=1}^n\text{exp}\{y_i\text{log}\frac{p_i}{1-p_i}+\text{log}(1-p_i)\}=\prod _{i=1}^n\{y_i({\mathbf{\text{d}}}_i\gamma +{\mathbf{x}}_i\beta )-\text{log}(1+e^{{\mathbf{d}}_i\gamma +{\mathbf{x}}_i\beta }\left)\right\},$$

where y = (y₁,…,y_n)^T. Subsequently, the log-likelihood function of (γ, β) becomes

$$\ell (\beta ,\gamma ;\mathbf{y},\mathbf{d},\mathbf{x})=\text{log}L(\gamma ,\beta ;\mathbf{y},\mathbf{d},\mathbf{x})=\sum _{i=1}^n\{y_i({\mathbf{d}}_i\gamma +{\mathbf{x}}_i\beta )-\text{log}(1+e^{{\mathbf{d}}_i\gamma +{\mathbf{x}}_i\beta })\}.$$ (1)

Sparse regression using ℓ₁-regularization is one of the most popular tools for high-dimensional data. The FLLR applied in this paper proposes minimizing

$$\sum _{i=1}^n\{-y_i({\mathbf{d}}_i\gamma +{\mathbf{x}}_i\beta )+\text{log}(1+\text{exp}({\mathbf{d}}_i\gamma +{\mathbf{x}}_i\beta )\left)\right\}+\mathrm{\Omega }(\beta ;{\lambda }_1,{\lambda }_2),$$ (2)

where

$$\mathrm{\Omega }(\beta ;{\lambda }_1,{\lambda }_2)={\lambda }_1\sum _{j=1}^t|{\beta }_j|+{\lambda }_2\sum _{j=2}^t|{\beta }_j-{\beta }_{j-1}|.$$

Note that penalties are only on β.

Lee et al. (2006) show that the problem of maximizing the log-likelihood function (1) can be reformulated as a weighted ordinary least squares problem by taking the 2nd order Taylor expansion. Following the steps described in Lee et al. (2006), the objective function (2) can be quadratically approximated. If the current estimate of γ and β are denoted γ̂^(r) and β̂^(r), respectively, the objective function is given by:

$$Q(\gamma ,\beta |{\hat{\gamma }}^{(r)},{\hat{\beta }}^{(r)})=\sum _{i=1}^n{w}_i{(z_i-{\mathbf{d}}_i\gamma -{\mathbf{x}}_i\beta )}^2+\mathrm{\Omega }(\beta ;{\lambda }_1,{\lambda }_2),$$ (3)

by letting

$$w_i={\hat{p}}_i(1-{\hat{p}}_i),z_i={\mathbf{d}}_i{\hat{\gamma }}^{(r)}+{\mathbf{x}}_i{\hat{\beta }}^{(r)}+\frac{y_i-{\hat{p}}_i}{{\hat{p}}_i(1-{\hat{p}}_i)}:\mathit{\text{adjusted response}}{\hat{p}}_i=g^{-1}({\mathbf{d}}_i{\hat{\gamma }}^{(r)}+{\mathbf{x}}_i{\hat{\beta }}^{(r)}).$$

Algorithms to solve the original fused lasso problem using a squared loss have been widely discussed in the literature. However, algorithms for logistic loss have rarely been discussed. In this paper, we consider a modified coordinate descent (CD) algorithm for the FLLR. The proposed algorithm can be considered as a logistic modification of the pathwise coordinate descent algorithm of Friedman et al. (2007). However, our procedure does not utilize the monotonicity (in fusion penalty) of the solution, which implies that, given λ₁, if coefficients of two variables are fused at λ₂ = c, then they are also fused for all λ₂ > c. As they have proved, monotonicity is true only for a few special cases of the covariate matrix.

Our CD algorithm also has “descent” and “fusion” steps. The algorithm iterates these two steps. The descent step is applied for β_k with β̂_j, j ≠ k, for each k. When the descent step fails to improve the objective function, the fusion step is applied, that is, we set β_k = β_k−1 = η (or β_k = β_k+1 = η) and do the descent step with respect to η. In other words, we move two parameters β_k and β_k−1 (or β_k+1) together. The detailed steps are as follows:

1. Let γ̂_j = 0 for j = 1, 2,…, s, β̂_k = 0, for k = 1, 2,…, t, and, for i = 1, 2,…, n, p̂_i = 1 /{ 1+exp (−d_iγ̂− x_i β̂)}, w_i = p̂_i (1−p̂_i), z_i = d_iγ̂+x_i β̂+ (y_i−p̂_i)/{p̂_i (1− p̂_i)}, $z_i^{*}=\sqrt{w_i}{z}_i,{\mathbf{d}}_i^{*}=\sqrt{w_i}{\mathbf{d}}_i$ and ${\mathbf{x}}_i^{*}=\sqrt{w_i}{\mathbf{x}}_i$.

2. (update of γ) We consider the derivative of (3) for γ_j, for j = 1, 2,…, s, given β = β̂, which is

   $$\frac{\partial Q(\gamma ,\hat{\beta }|\hat{\gamma },\hat{\beta })}{\partial {\gamma }_j}=\sum _{i=1}^n\{2{(d_{ij}^{*})}^2{\gamma }_j-d_{ij}^{*}\{z_i^{*}-\sum _{l\ne j}{d}_{il}^{*}{\gamma }_l-\sum _{k=1}^t{x}_{ik}^{*}{\hat{\beta }}_k\left\}\right\},\text{for}j=1,2,\dots ,s.$$ (4)

   We update γ̂ by solving the above set of normal equations by setting (4) equal to 0.

3. (descent step) In this step, we consider the derivative of (3) for β_k, given β_l = β̂_l, l ≠ k, which is

   $$\frac{\partial Q(\hat{\gamma },\beta |\hat{\gamma },\hat{\beta })}{\partial {\beta }_k}=\sum _{i=1}^n\{2{(x_{ik}^{*})}^2{\beta }_k-x_{ik}^{*}(z_i^{*}-\sum _{j=1}^s{d}_{ij}^{*}{\hat{\gamma }}_j-\sum _{l\ne k}{x}_{il}^{*}{\hat{\beta }}_l)\}+{\lambda }_1\mathrm{s}\mathrm{g}\mathrm{n}({\beta }_k)-{\lambda }_2\mathrm{s}\mathrm{g}\mathrm{n}({\hat{\beta }}_{k+1}-{\beta }_k)+{\lambda }_2\mathrm{s}\mathrm{g}\mathrm{n}({\beta }_k-{\hat{\beta }}_{k-1}),$$ (5)

   where sgn(x) is the subgradient of |x|. The function (5) is piecewise linear with breaks at {0, β̂_k−1, β̂_k+1} and let β̃_k be the solution to ∂Q(β_k) / ∂β_k = 0. If Q(β̃_k) ≤ Q(β̂_k), then let β̂_k = β̃_k and do the descent step to update β_k+1. If not, we do the fusion step below to update β_k and β_k−1, simultaneously.
4. (fusion step) In this step, we move β_k and β_k−1 together. We set η = β_k = β_k−1. The derivative of (3) with respect to η for given β_j = β̂_j, j ≠ k, (k − 1) is

   $$\sum _{i=1}^n\{2{(x_{i(k-1)}^{*}+x_{ik}^{*})}^2\eta -(x_{i(k-1)}^{*}+x_{ik}^{*}\left)\right(z_i^{*}-\sum _{j=1}^s{d}_{ij}^{*}{\hat{\gamma }}_j-\sum _{l\ne k,(k-1)}{x}_{il}^{*}{\hat{\beta }}_l\left)\right\}+2{\lambda }_1\mathrm{s}\mathrm{g}\mathrm{n}(\eta )-{\lambda }_2\mathrm{s}\mathrm{g}\mathrm{n}({\hat{\beta }}_{k+1}-\eta )+{\lambda }_2\mathrm{s}\mathrm{g}\mathrm{n}(\eta -{\hat{\beta }}_{k-2}).$$ (6)

   The same procedure in the descent step finds the solution, say η̃, to (6) = 0. Update β_k−1 and β_k with η̃, and go to the descent step for β_k+1.

5. We repeat the descent and fusion steps until convergence.

6. For i = 1,…, n, update p̂_i, w_i, $z_i^{*}$, ${\mathbf{d}}_i^{*}$, and ${\mathbf{x}}_i^{*}$.

7. Repeat steps 2–6 until convergence.

Generally, difficulties in the fused lasso regularized model arise in the computation speed and finding the optimal tuning parameters. When the number of features and samples are very large, say, over 100,000 features with 200 samples, the algorithm needs to be improved to solve the problem efficiently. The choice of tuning parameters is critical since tuning parameters determine the number of non-zero coefficients. For example, if tuning parameters are small enough, then many coefficients will be zero. Several criteria such as the AIC and the BIC are suggested for finding the optimal tuning parameters. In this study, since we are interested in building a predictive model of logistic regression, tuning parameters (λ₁, λ₂) are chosen to minimize the predictive (Pearson) residual sum of squares, that is, for a new observation (y, d, x),

$$\mathrm{P}\mathrm{R}(\lambda )=\mathrm{E}\left\{\frac{{(y-{\hat{p}}_{\lambda }))}^2}{{\hat{p}}_{\lambda }(1-{\hat{p}}_{\lambda })}\right\},$$ (7)

where p̂_λ = 1 /{1 + exp (− dγ̂_λ − xβ̂_λ)}, and γ̂_λ and β̂_λ are the estimates of γ and β with tuning parameters λ = (λ₁, λ₂).

We estimate the above predictive residual using a q-fold cross-validation (CV) method in order to choose the tuning parameters in an adaptive way. To be specific, let {S_k : k = 1,…, q} be a partition of all samples {(y₁, d₁, x₁), (y₂, d₂, x₂),…, (y_n, d_n, x_n)}, whose size of S_k is approximately equal to [n/q]. For each k = 1,…, q, let ${\hat{\gamma }}_{\lambda }^{(-k)}$ and ${\hat{\beta }}_{\lambda }^{(-k)}$ be estimates from a training set {S_j : j ≠ k} with λ = (λ₁, λ₂). Then, the predictive residual is estimated by

$$\widehat{\mathrm{P}\mathrm{R}}(\lambda )=\sum _{k=1}^q\sum _{i\in S_k}\frac{{(y_i-{\hat{p}}_{\lambda ,i}^{(-k)})}^2}{{\hat{p}}_{\lambda ,i}^{(-k)}(1-{\hat{p}}_{\lambda ,i}^{(-k)})},$$

where ${\hat{p}}_{\lambda ,i}^{(-k)}=1/\{1+\text{exp}(-{\mathbf{d}}_i{\hat{\gamma }}_{\lambda }^{(-k)}-{\mathbf{x}}_i{\hat{\beta }}_{\lambda }^{(-k)}\left)\right\}$. We perform a grid search to find the minimum point of $\widehat{\mathrm{P}\mathrm{R}}(\lambda )$.

3. Results

3.1. Preliminary analysis

It is known that both intra-cranial volume (ICV) and CC cross-sectional areas decrease with age (Holloway et al., 1993). Larger ICV also correlates with larger CC (Bruner et al., 2012). Also, there is a potential of sexual dimorphism in the CC (Ardekani et al., 2012b). Therefore, it is necessary to remove the effects of age, ICV and sex, which are considered as potential confounding effects, in examining the callosal differences for AD. In OASIS, the ICV is estimated by the eTIV (estimated total intra-cranial volume) variable and distributed along with the imaging data. Figure 2 shows the effects of age, ICV (represented by eTIV^1/3) and sex on the CC thickness. Figure 2(a) shows the mean curves of raw CC thickness for each group while Figure 2(b) shows the callosal differences of NC and AD from the mean after removing the effects of those variables. To obtain the thickness profiles of Figure 2(b), we take the residuals from the regression model of callosal thickness against age, ICV and sex at each location, then take the average of the residuals for each group at each location. Figure 2(b) shows a general thinning of the CC in AD as well as greater than average thinning in the areas of the genu and splenium.

Figure 2.

(a) Mean of raw callosal thickness profiles in NC (solid line) and AD (dotted line). (b) Mean adjusted callosal thickness profiles for each group after removing the effects of age, ICV and sex.

It is natural to consider the ANCOVA model with covariates of age, sex and ICV to test the callosal difference between NC and AD at each location, i.e., for each j = 1, …, 99,

$$x_{ij}={\alpha }_0+{\alpha }_1\mathrm{A}\mathrm{G}{\mathrm{E}}_i+{\alpha }_2\mathrm{S}\mathrm{E}{\mathrm{X}}_i+{\alpha }_3\mathrm{e}\mathrm{T}\mathrm{I}{\mathrm{V}}_i^{1/3}+{\alpha }_4\mathrm{A}{\mathrm{D}}_i+{\epsilon }_i$$

where SEX and AD are the indicator functions for female and AD, respectively, and ε_i follows the standard normal distribution. Figure 3(a) presents the p-values from the ANCOVA indicating that most locations show significance at the level of 0.05 without any correction for multiple testing, which is not very informative in showing regional specific differences for predictive differentiation between AD and NC.

Figure 3.

(a) The p-values obtained from ANCOVA at each location. The p-values are below 0.05 at most locations on CC. (b) Estimated coefficients from the proposed FLLR method. Non-zero coefficients are found at the locations of 1–9, 18–23, 34–36, 88–90.

3.2. FLLR analysis

Next we applied our proposed model to callosal thickness profiles after removing the potential confounding effects of age, sex and ICV. OASIS also provides a measure for cognitive impairment, the Mini-Mental State Examination (MMSE) score. MMSE is a simple questionnaire test that is widely used by clinicians to screen for dementia. The range of MMSE score is from 0 (worst) to 30 (best). To utilize all the information available, we include MMSE in our model, which might increase the prediction power of the model. If other biomarkers such as CSF beta-amyloid and tau protein levels are available, they can also be incorporated in the model. Thus, we generate a classifier by combining our callosal thicknesses with subject’s MMSE scores as follows:

$$\log \frac{p_i}{1-p_i}={\gamma }_0+{\gamma }_1\mathrm{M}\mathrm{M}\mathrm{S}\mathrm{E}+\sum _{j=1}^{99}{\beta }_j{\tilde{x}}_{ij},$$ (8)

with the fused lasso penalty of ${\lambda }_1{\sum }_{j=1}^{99}|{\beta }_j|+{\lambda }_2{\sum }_{j=2}^{99}|{\beta }_j-{\beta }_{j-1}|$ where x̃_ijs are residuals from the regression model, i.e., ${\tilde{x}}_{ij}=x_{ij}-({\hat{\alpha }}_0+{\hat{\alpha }}_1\mathrm{A}\mathrm{G}{\mathrm{E}}_i+{\hat{\alpha }}_2\mathrm{e}\mathrm{T}\mathrm{I}{\mathrm{V}}_i^{1/3}+{\hat{\alpha }}_3\mathrm{S}\mathrm{E}{\mathrm{X}}_i)$, and α̂₀, α̂₁, α̂₂, α̂₃ are estimates by the ordinary least squares method. The tuning parameters (λ₁, λ₂) are chosen using 5-fold CV as described in Section 2.4 in the range of (λ₁, λ₂) ∈ [1.0, 1.0] × [5.0, 5.0] by interval 0.1. The computation time for the grid search was approximately two hours on a 2.9 GHz Quad-core Linux computer. In practice, one can start with arbitrarily large λ₁ to preserve most coefficients, followed by finding the optimal λ₂. Then, decrease λ₁ slightly and repeat the search for λ₂ in reasonable range around the previous optimal value of λ₂, and repeat the procedure until the minimum is reached.

Figure 3(b) shows the estimated coefficients β̂_is of the proposed model. We found sparse but contiguous regions that indicate possible callosal differences where β̂_i ≠ 0. Regions [1–9], [18–23], [36–40] and [88–90] have negative coefficients suggesting that these regions are narrower in AD than in NC, and differentiate patients with early AD from NC. Table 2 reports the classification error rate based on our proposed method using 5-fold CV method. The accuracy of our classifier was 0.84 (165/196) with 78% sensitivity and 91% specificity.

Table 2.

Classification error rate by 5-fold cross-validation: accuracy=0.84(=165/196) with 78% sensitivity and 91% specificity. Out of 22 false negatives, 20 subjects with CDR=0.5 and 2 subjects with CDR=1 were misclassified as normal.

True	NC	AD
Est.
NC	89	22
AD	9	76
Total	98	98

4. Discussion and conclusion

In this paper, we have proposed using FLLR (fused lasso logistic regression) to find spatially correlated regions of the CC where thickness may be predictive of AD. Analysis of callosal thickness profiles provide a more detailed view of the differences between AD and NC groups than can be provided by using CC subdivisions. The proposed approach has several advantages in comparison with subdivision procedures. Subdivision procedures are usually done in two steps: subdivisions are determined either geometrically (e.g., splenium as the posterior 1/5 of the CC) or by factor analysis; and equality of the subdivision areas between the two populations is tested. In this approach, CC subdivisions are chosen subjectively or by noisy factor loadings which can be unreliable. The proposed FLLR method reformulates the problem as a feature selection problem in the framework of a prediction model, and finds differential subregions adaptively. For this reason, the FLLR method finds differential subregions between two populations more objectively and reliably. Our method is more specific than analysis based on subdivisions of the CC since we consider entire callosal thickness profiles and take neighborhoods into account in our model. In addition, it simultaneously provides a predictive model which may be valuable for diagnosis.

Finally, we have applied the proposed FLLR to find subregions of the CC where thicknesses differentiate early AD from NC. Our proposed method found that sections of the genu and splenium of AD are proportionally thinner than those of NC with 84% prediction accuracy. Nine normal subjects were misclassified as patients. In clinical practice, such subjects could be candidates for further evaluation. Twenty-two patients with AD were misclassified as normal. Of the 22 false negatives, 20 subjects with CDR=0.5 and 2 subjects with CDR=1 were misclassified as normal. A possibility of misdiagnosis should be considered further in these cases since thinning of the CC appears in the normal range incongruent with there clinical symptoms. A careful study with more extensive follow up is required to determine if these misclassified subjects of the CDR=0.5 group have clinical significance.

The regions identified by this method include parts of the genu that are connected to the frontal lobes, and a region in the splenium that seems to contain fibers that run to the medial temporal lobes (Hofer and Frahm, 2006). Both are regions that are implicated in AD. The region in the splenium seems to have a lesser weight, possibly due to its being a narrow tract whose position suffers from variability.

The classifier we have described is based on two commonly used diagnostic procedures, the MMSE and a structural MRI. Our automatic analysis programs require little supervision or intervention. The MRI protocol involved does not require long or complex acquisitions. The CC can be segmented in a fast, automatic, and reliable way. The FLLR model can be easily extended to include other brain or chemical biomarkers of AD.